What is the derivative of f(t) = (tcos^2t , t^2sin^2t-cost ) ?

Mar 23, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{t}^{2} \cos t + 2 t \cos t + \sin t}{{\cos}^{2} t + 2 t \sin t \cos t}$

Explanation:

In parametric form of equation $f \left(t\right) = \left(x \left(t\right) , y \left(t\right)\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}$

Here $x \left(t\right) = t {\cos}^{2} t$ and $y = {t}^{2} \sin t - \cos t$

and therefore $\frac{\mathrm{dx}}{\mathrm{dt}} = {\cos}^{2} t - t \times 2 \cos t \times \left(- \sin t\right)$

= ${\cos}^{2} t + 2 t \sin t \cos t$

and $\frac{\mathrm{dy}}{\mathrm{dt}} = {t}^{2} \cos t + 2 t \cos t + \sin t$

Hence $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{t}^{2} \cos t + 2 t \cos t + \sin t}{{\cos}^{2} t + 2 t \sin t \cos t}$